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Validity of the Particle in a 1-D box for Conjugated Polyynes

R. Viswanathan

The particle in a one-dimensional box model is a simple quantum mechanical model that can be used to predict the electronic energy levels for the pi electrons in long-chain conjugated unsaturated compounds. This exercise compares the results predicted by the particle-in-a-box model with semi-empirical quantum mechanical calculations.

According to the particle in a 1-D box model, the energy of the electron in a quantum state n is given by:

En = n2h2/8ml2

where h is the Planck's constant, m is the mass of the electron, and l is the length of polyyne. The length of the box is estimated using the C-C triple, single, and C-H bond lengths. The number of pi electrons is 4 times the number of triple bonds in the conjugated polyyne. The HOMO (n) is one half the number of pi electrons. The energy change corresponding to the HOMO-LUMO transition is given by:

DE = En+1 - En

The wavelength corresponding to the transition is calculated as:

lmax = hc/DE


Compound

No. of carbon atoms

Length of chain

lmax 1D-box

lmax MOPAC


1,3-butadiyne 4 5.87 126 196
1,3,5-hexatriyne 6 8.43 180 210
1,3,5,7-octatetrayne 8 10.99 234 192
1,3,5,7,9-decapentayne 10 13.55 288 204
1,3,5,7,9,11-dodecahexayne 12 16.11 342 220

The structures of these polyyne molecules were minimized using the molecular mechanics method, followed by a semi-empirical method. The wavelength corresponding to the HOMO-LUMO transition were then calculated using ZINDO. The values for the wavelength of maximum absorption obtained by the semi-empirical method are compared with the particle in a box results in the following graph.

Both models predict an increase in the wavelength with increasing chain length. The effect predicted by the particle-in-a-box model is more pronounced than the results obtained using MOPAC.

Viswanathan